Problem: All of the 5th grade teachers and students from Santa Rita went on a field trip to an art museum. Tickets were $$5.50$ each for teachers and $$2.00$ each for students, and the group paid $$40.00$ in total. The next month, the same group visited a natural history museum where the tickets cost $$22.00$ each for teachers and $$7.50$ each for students, and the group paid $$155.50$ in total. Find the number of teachers and students on the field trips.
Answer: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5.5x+2y = 40}$ ${22x+7.5y = 155.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-22x-8y = -160}$ ${22x+7.5y = 155.5}$ Add the top and bottom equations together. $ -0.5y = -4.5 $ $ y = \dfrac{-4.5}{-0.5}$ ${y = 9}$ Now that you know ${y = 9}$ , plug it back into $ {5.5x+2y = 40}$ to find $x$ ${5.5x + 2}{(9)}{= 40}$ $5.5x+18 = 40$ $5.5x = 22$ $x = \dfrac{22}{5.5}$ ${x = 4}$ You can also plug ${y = 9}$ into $ {22x+7.5y = 155.5}$ and get the same answer for $x$ ${22x + 7.5}{(9)}{= 155.5}$ ${x = 4}$ There were $4$ teachers and $9$ students on the field trips.